0

Let $I_{3}$ be an inverse semigroup consisting of all partial bijections on a set $\{1,2,3\}$, called the symmetric inverse semigroup. Then \begin{align*} I_{3}=\left\{\emptyset, \binom{1}{1}，\binom{1}{2}，\binom{1}{3}，\binom{2}{1}，\binom{2}{2}，\binom{2}{3}，\binom{3}{1}，\binom{3}{2}，\binom{3}{3}，\binom{1,2}{1,2}，\\ \binom{1,2}{1,3}， \binom{1,2}{2,3}，\binom{1,2}{2,1}，\binom{1,2}{3,1}，\binom{1,2}{3,2}，\binom{1,3}{1,2}, \binom{1,3}{1,3}, \binom{1,3}{2,3}, \binom{1,3}{2,1}, \binom{1,3}{3,1},\\ \binom{1,3}{3,2}, \binom{2,3}{2,3}, \binom{2,3}{2,1}, \binom{2,3}{3,1}, \binom{2,3}{3,2}, \binom{2,3}{1,2}, \binom{2,3}{1,3}, \binom{1,2,3}{1,2,3}, \binom{1,2,3}{1,3,2}, \\ \binom{1,2,3}{2,3,1}, \binom{1,2,3}{2,1,3}, \binom{1,2,3}{3,2,1}, \binom{1,2,3}{3,2,1}\right\}. \end{align*}

I want to find all generators and relations of $I_{3}$. Can someone help me? Thanks in advance.

bing
  • 1,160
  • 1
    What do you mean by all generators and relations? In particular, what does "all" mean in this context? I_3 is generated by any subset that contains a generating set for the symmetric group S_3 and any partial permutation defined on only 2 points. – James Mitchell Mar 29 '16 at 19:56
  • @James Thank you very much. I want to try to find generators and relations of $I_{3}$ as possible. There are other generators and relations except you listed? – bing Mar 30 '16 at 03:49
  • 1
    The only generating sets are the ones I mentioned in my previous comment. I don't that trying to find all relations is a meaningful question, for every generating set there are infinitely many presentations. Could you be more precise in what you are looking for and why? – James Mitchell Apr 03 '16 at 18:31
  • @ James Mitchell I just have this idea, this problem is interesting. – bing Apr 15 '16 at 13:00

0 Answers0